the maximum likelihood estimate \(p_n^*\)  (Equation REF ).

For statistical significance, the entire sampling process was repeated \(1{,}000\)  times independently, all our graphs include 90% confidence bounds over these independent runs.
Furthermore, to investigate how sensitive our estimators are to misestimated \(\hat{\lambda }\)  values, we computed the estimator values with both the actual \(\lambda \)  values and estimated \(\hat{\lambda }\)  values.
We followed the estimation scheme of Section REF , generating additional ratings with \(K=1\) , \(p_1=0.4\) , \(M=20\)  and \(n=100\)  to represent a plausible production-level scenario [1]}.
The result obtained from fitting a curve: \(\hat{\lambda }_i = a+(1-a)\cdot b^{i-1}\)  where \(a \in [0,1]\)  and \(b \in (0,1)\)  to the data was: \(\hat{\lambda }_i=0.33+0.67\cdot 0.92^{i-1}\) .
We only report the results for misestimated \(\hat{\lambda }\)  values in the strong bandwagon setting, due to high similarity with the weak setting.
Estimator Bias and Convergence
<FIGURE><FIGURE>Our results are visualized in Figure REF ,
which displays the convergence rate of the sample mean under various levels of bandwagon effect, and Figure REF , which shows the behavior of all estimators with known \(\lambda \)  in the weak and strong bandwagon settings.
First, we consider the influence of the bandwagon effect on the sample mean (\(\bar{p}_n\) ):
from Figure REF  it is immediately clear that the effect's strength has a significant impact on the rate of convergence.
Whilst under no bandwagon effect the confidence intervals are within \(0.05\)  of \(p\)  after fewer than 300 samples and within \(0.01\)  after roughly \(8{,}000\)  samples, under the weak bandwagon effect these numbers exceed \(1{,}000\)  and \(33{,}000\)  respectively.
Under the strong bandwagon effect the confidence intervals increase even further, exceeding \(0.05\)  after \(10^6\)  samples.
However, in all cases the confidence bounds still appear to decrease steadily, suggesting that \(\bar{p}_n\)  will converge eventually, as also predicted by the sufficient condition for consistency (Equation REF ).
Nevertheless, it is also clear that the bandwagon effect introduces a significant loss in data efficiency.
In practice, this could mean that an infeasibly large amount of data is required to get a decent estimate under a strong bandwagon effect.
Next, we examine the impact on the bias of the sample mean and other estimators.
As shown in Figure REF  (top, middle), even prior to convergence, all estimators including \(\bar{p}_n\)  make negligible errors in expectation, strongly suggesting that they are all unbiased.
Whilst the unbiasedness was already theoretically proven for \(\bar{p}_n\)  and the affine estimators \(\hat{p}_n\)  and \(\tilde{p}_n\) , it is surprising to see that even at low numbers of samples the expected error of the maximum likelihood estimate \(p_n^*\)  appears negligible.
Our results thus seem to indicate that all the estimators are unbiased or have an extremely small bias, further confirming that the bandwagon effect is not a bias problem.
We then focus on the weak bandwagon condition (Figure REF  top).
Here, all estimators converge to \(p\) , at a rate similar to that of \(\bar{p}_n\) .
That the sample mean and affine estimators converge is not surprising, as the setting meets the sufficient condition for consistency.
However, in this scenario none of the estimators actually provide a clear improvement over the sample mean, indicating that they are unable to apply significant beneficial corrections to the bandwagon effect.
The lack of improvement over \(\bar{p}_n\)  is particularly concerning as this indicates that our estimators are similarly affected by the loss of data efficiency induced by the bandwagon effect.
Nonetheless, one may expect the convergence to slow down even further in the strong bandwagon setting shown in the middle row of Figure REF .
Here, both the affine and maximum likelihood estimators are actually able to converge on the true value much sooner than the sample mean.
However, confidence bounds of the former also become wider after roughly 10 sampled ratings and stay wider until roughly \(1{,}000\)  samples, with the weighting of \(\tilde{p}_n\)  seemingly reducing this effect when compared with \(\hat{p}_n\) .
Thus during this period, the sample mean actually appears to be a more reliable choice in terms of expected error.
This observation illustrates that choosing between estimators is not always straightforward, as in this case the optimal choice between \(\bar{p}_n\)  and \(\tilde{p}_n\)  depends on the number of samples that are available.
Importantly, the maximum likelihood estimator does not have this drawback: its confidence bounds never appear wider than the sample mean while still converging quicker.
Across both settings, we have thus observed no noticeable bias problems for any estimator.
Importantly, the sample mean has serious convergence issues under the strong bandwagon, where the other estimators are able to converge much sooner.
Nonetheless, despite its high efficiency, the affine estimator was also found to initially have the highest variance.
It appears that when \(\lambda \)  values are known, the maximum likelihood estimator is the most reliable in spite of the lack of proven theoretical guarantees for this estimator.
However, even the maximum likelihood estimator is unable to fully undo the slowdown introduced by either level of the bandwagon effect, highlighting the substantial impact this effect still has on effective sample size and relevance estimation.

Bandwagon Misestimation
To investigate the effect of misestimated \(\hat{\lambda }\)  values, we look at the bottom row in Figure REF .
In particular, it shows what happens when we underestimate a strong bandwagon effect.
Clearly, we see that no noticeable bias has been introduced to any estimator.
Interestingly, the affine estimators actually suffer less from initial variance:
however, we speculate that overestimating the bandwagon effect could instead enlarge the variance. Nevertheless, all our estimators converge substantially slower than when \(\lambda \)  was known (cf. middle row), with their confidence bounds now only moderately smaller than those of \(\bar{p}_n\) .
We thus conclude that the ability of our estimators to accelerate convergence is very sensitive to correct estimation of \(\lambda \) , where we do not observe an advantage for any of the estimators.
The results with misestimated \(\hat{\lambda }\)  values demonstrate that \(\lambda \)  estimation is a crucial part of effective bandwagon correction.
Researching bandwagon estimation methods is therefore an important task for future work: since without robust estimation, correcting for the bandwagon effect appears to remain an open problem.

Discussion
So far, we have investigated the bandwagon effect in both theoretical and empirical terms.
In contrast with the well-studied selection bias [2]}, [3]}, [4]}, [5]}, our experimental results indicate that the bandwagon effect mainly poses a problem for convergence without introducing noticeable statistical bias.
Yet
the effect is generally referred to as a bias by previous work [6]}, [7]}, [8]}, [9]}.
This raises a higher-level conceptual question: should the bandwagon effect be interpreted as a bias?
The remainder of this section will contrast three possible perspectives on this question that we deem valuable for the understanding of the field.
On one hand, both our theoretical and empirical results clearly indicate that statistical bias is not the issue.
The expected value of both the sample mean and individual ratings is equal to the true relevance, with
no noticeable error in the mean value of each estimator in our experiments.
Therefore, in statistical terms, it seems that we are purely dealing with convergence and consistency issues and not with bias.
This distinction is important as solutions to inconsistency may be very different than those for bias: i.e. our experimental results suggest that methods based on debiasing techniques are not particularly effective at solving convergence issues.
On the other hand, our theoretical analysis also revealed that the bandwagon effect introduces conditional bias:
given that there is an error in the sample mean after \(m\)  ratings, the expected value of any subsequent rating or sample mean will be incorrect.
In other words, under the bandwagon effect any errors in the sample mean, which are unavoidable in practice, will subsequently introduce (conditional) bias.
A proponent of the first perspective could argue that this view is mistaking convergence issues for bias.
Nevertheless, we argue that as long as both parties understand the difference between bias and conditional bias, their disagreement is mainly about semantics and not on the underlying nature of the issue.
Alternatively, one could consider the bandwagon effect as actually changing the true preferences of users.
This view poses that the observed ratings are not a distortion of the true preferences but that the actual preferences of users have changed.
One could argue that this is in line with the users' experience as they do not make a conscious choice to alter their ratings in response to others' feedback but aim to assign ratings according to their satisfaction.
Instead, their enjoyment of a product may still be subconsciously influenced by the enjoyment of others.
Thus the advantage of this view is the apparent alignment with the users' own experience. However, their experience is precisely unaware of any subconscious factors such as the bandwagon effect.
The danger is that by not viewing the bandwagon effect as a misrepresentation of preferences, one may be less inclined to correct for it.
However, when training a recommender system, it surely seems critical to know whether high ratings were given because a user was affected by previous users, or whether they made this decision solely based on the item itself.
Apart from these conceptual perspectives, we strongly argue that in statistical terms the bandwagon effect is a problem of convergence and consistency and not of statistical bias.
Nevertheless, outside of pure statistical terms, we think that each of the three views brings a valuable perspective to the field.
Most importantly, we think the field should be aware of the difference between the bandwagon effect and statistical bias.
As long as this distinction is understood, we think it is acceptable if one chooses to conceptually categorize the bandwagon effect as another form of bias, although we would advise against it. Overall, our work aims to promote a more complete statistical understanding of the issues surrounding user interactions that goes beyond bias to include variance, consistency and convergence.

Conclusion
In this paper we examined the bandwagon effect: an underexplored phenomenon where users' interactions are affected by interactions of earlier users.
While previous work has approached the bandwagon effect as problem of bias [6]}, [7]}, our theoretical analysis revealed that it does not add bias to the rating distribution nor to the sample mean.
Instead, we have shown that it is better viewed as a problem of convergence and consistency. We derived a necessary condition for consistency under the effect, thereby also adding to earlier consistency claims [6]}.
We then proposed several novel estimators to mitigate the convergence issues.
While our empirical analysis revealed that our novel estimators can substantially increase the convergence rate under a strong bandwagon effect, all estimators also carry their own limitations.
Moreover, even when using the empirically best-performing estimator, our
purposely simple model of the bandwagon effect still significantly
increases the number of samples required for accurate relevance estimation.
Future work may wish to examine the impact of more complex bandwagon models (e.g. [13]}, [8]}) as well as parameter and rating scale choices on bias and convergence, both in unpersonalized as well as personalized settings.
For the latter case in particular, user preference may be further associated with the time of arrival (e.g. item's fans are both more likely to like it and to rate it early) or \(\lambda \)  values for one timestep may vary across users.
Nevertheless, even under our simple model, our findings lead us to conclude that there is no clear and well-understood solution for the bandwagon effect.
Acknowledgments
This work was supported by the Google Research Scholar Program.
All content represents the opinion of the author, which is not necessarily shared or endorsed by their employers and/or sponsors.

Code and data
To facilitate the reproducibility of the reported results, our experimental implementation is publicly available at https://github.com/NKNY/bandwagonictir2022.

Appendix
[Bias of sample mean and single rating]

\(\bar{p}_n\)  and \(r_n\)  (Equation REF ) are unbiased estimators of parameter \(p\)  for all \(n\) :
\(\mathbb {\forall }n: \mathbb {E}\left[\bar{p}_n\right] = \mathbb {E}\left[r_n\right] = p.\) 


\(\mathbb {E}[\bar{p}_1]=\mathbb {E}[r_1]=p\)  by Equations REF -REF . Then by induction:

E[pn]=
(n-1)E[pn-1]+E[rn]n
=n p + (n-n)E[pn-1]n=p.
\(r_n\)  is unbiased as it is a weighted sum of \(p\)  and unbiased \(\mathbb {E}[\bar{p}_{n-1}]\) :

E[rn]
= E[n p + (1-n)pn-1]
= n p + (1-n)E[pn-1]
= p.    
[Variance of \(r_i\) ]

Under the bandwagon effect the variance of any rating \(r_i\)  is \(p(1-p)\) .
By Theorem :

i: V[ri]
=E[ri2]-E2[ri]
=E[ri]-E2[ri]=p-p2.

 []

Covariance of \(\bar{p}_{n-1}\)  and \(r_n\)  is \((1-\lambda _n)^2\mathbb {V}[\bar{p}_{n-1}]\) .
Cov [pn-1, rn]
= E[(pn-1-p)(n p + (1-n)pn-1-p)]
= (1-n)2V[pn-1]. 
 [Efficiency of sample mean]

The efficiency of the sample mean under the bandwagon effect defined in Equation REF  is
\(\mathbb {E}[(\bar{p}_n-p)^2]=p(1-p) \bigg (\dfrac{1}{n^2} + \sum _{i=1}^{n-1}\dfrac{1}{i^2} \prod _{j=i+1}^{n} \dfrac{(j-1)(j+1-2\lambda _j)}{j^2}\bigg ).\) 


By Theorems  and Lemmas  , :

E[(pn-p)2]
= (n-1)2n2V[pn-1]+2(n-1)n2Cov[rn,pn-1]
    +V[rn]n2 =(n-1)(n+1-2n)n2V[pn-1]+p(1-p)n2.
 The proof is completed by unrolling the process to \(\mathbb {V}[\bar{p}_{1}]=p(1-p)\)  and collecting all the terms. []
The asymptotic effiency of the sample mean is:
\(\lim _{n \rightarrow \infty } \mathbb {E}[(\bar{p}_n-p)^2]= \lim _{n \rightarrow \infty } p(1-p) \bigg (\sum _{i=1}^{n-1}\dfrac{1}{i (i+1)}\prod _{j=i+1}^{n-1}\bigg (1-\dfrac{2\lambda _j}{j+1}\bigg )\bigg ).\) 


By re-grouping numerators and denominators in the product of Theorem  and cancelling out extra terms:

E[(pn-p)2]
=p(1-p) (1n2 + i=1n-1
n+1-2ni (i+1)nj=i+1n-1
(1-2jj+1) ).

The proof is completed by taking \(\lim _{n\rightarrow \infty }\)  of above expression.

[Sample mean consistency]

\(\sum \lambda _i^2\)  diverging is a necessary condition for the consistency of \(\bar{p}_n\)  (Equation REF ):
\(\forall \epsilon >0: \lim _{n \rightarrow \infty } \text{Pr}(|\bar{p}_n-p|>\epsilon )=0 \longrightarrow \lim _{n \rightarrow \infty } \sum _{i=1}^n \lambda _i^2 = \infty .\) 


By [15]}, consistency of \(\bar{p}\)  implies its variance vanishing: \(\forall \epsilon >0: \text{Pr}(|\bar{p}_n-p|\ge \epsilon )\ge \mathbb {V}\left[\bar{p}_n\right]-\epsilon ^2\) .
Then:
\(\forall \epsilon >0: \lim _{n \rightarrow \infty } \text{Pr}\left(\left|\bar{p}_n-p\right|>\epsilon \right)=0\rightarrow \lim _{n \rightarrow \infty } \text{Var}(\bar{p}_n)=0\rightarrow \forall i: \lim _{n \rightarrow \infty } \prod _{j=i+1}^{n-1}(1-\frac{2\lambda _j}{j+1})=0\rightarrow \forall i: \lim _{n \rightarrow \infty } \sum _{j=i+1}^{n-1}\frac{\lambda _j}{j+1}=\infty \rightarrow \forall i:\lim _{n\rightarrow \infty } \sqrt{\sum _{j=i+1}^{n-1}\lambda _j^2}\sqrt{\sum _{j=i+1}^{n-1}\frac{1}{\left(j+1\right)^2}}=\infty \rightarrow \lim _{n \rightarrow \infty } \sum _{i=1}^n \lambda _i^2 = \infty \) .
The second step is due to partial products in Theorem , the third by [16]}, the fourth by Cauchy-Schwarz inequality and the fifth by convergence of p-series with \(p=2\) .
[Conditional bias of sample mean]

 
\(\mathbb {E}[\bar{p}_n-p\mid \bar{p}_m]=(\bar{p}_m-p)\prod _{i=m+1}^n(1-\frac{\lambda _i}{i})\)  where \(m<n\) .


By repeatedly applying the induction step in Theorem  , replacing \(\mathbb {E}[\bar{p}_n]\)  with \(\mathbb {E}[\bar{p}_n\mid \bar{p}_m]\)  and noting that \(\mathbb {E}[\bar{p}_m|\bar{p}_m]=\bar{p}_m\) :
\(\mathbb {E}[\bar{p}_n-p\,|\,\bar{p}_m]= \mathbb {E}[\bar{p}_{n-1}-p\,|\,\bar{p}_m]\big (1-\frac{\lambda _n}{n}\big )= \big (\bar{p}_{m}-p\big )\hspace{-2.84526pt}\prod _{i=m+1}^n\hspace{-2.84526pt}\big (1-\frac{\lambda _i}{i}\big ).\) 
 [Error bound of \(\bar{p}_n\) ]

If \(\lambda _i=c^{i-1}\)  s.t. \(c \in (0,1)\) , the asymptotic expected absolute error of the sample mean is at least \(2p( 1-p) \exp ( -c\phi ( c,1,2)-c^{2}\phi ( c^{2},2,2) )\) , where \(\phi ( z,s,\)  \(a) \equiv \sum ^{\infty }_{k=0}\frac{z^{k}}{( a+k)^s }\)  is Lerch transcendent and converges to a fixed value \(\forall z<1\)  [17]}.



n  E[|pn-p|]
n  E[|E[pn-p|r1]|]
= 2p(1-p)
(i=2(1-ii))
2p(1-p)(i=2(-ii-i2i2))
=2p(1-p) ( -ci=0ci( 2+i) 
-c2i=0c2i( 2+i)2 ).

The first step is due to the law of total expectation and Jensen's inequality for conditional expectation. The second is by Proposition  with \(m=1\) . The third is due to a known inequality \(\forall x> -0.68: \ln (1+x) \ge x-x^2\)  [18]}.
 []

Each \(\hat{r}_{i}\)  (and thus also \(\hat{p}_n\) ) is unbiased for all \(\hat{\lambda }_i\) .


By Theorem  , for any choice of \(i\)  and \(\hat{\lambda }_i\) :
\(\mathbb {E}[\hat{r}_{i} \mid \hat{\lambda }_i]= \mathbb {E}\bigg [ \dfrac{r_i-(1-\hat{\lambda }_i)\bar{p}_{i-1}}{\hat{\lambda }_i}\bigg ]= \dfrac{\mathbb {E}[ r_i-\bar{p}_{i-1} ]}{\hat{\lambda }_i} + \mathbb {E}[ \bar{p}_{i-1} ]= p.\) 
 [Conditional bias of \(\hat{r}_{i}\mid \hat{\lambda }_i=\lambda _i\) ]

If \(\hat{\lambda }_i=\lambda _i\) , the affine mean estimator \(\hat{r}_{i}\)  is an unbiased estimator of \(p\)  when conditioned on \(\hat{r}_j\)  or \(\bar{p}_{j}\)  s.t. \(\forall j<i\) .


For any function of previous ratings \(f_{i}=f(r_1, \ldots , r_{i-1})\) :

E[rifi,i] = E[i p + (1-i)pi-1-(1-i)pi-1ifi]
=iip+(i-ii)E[pi-1fi] = p.


The proof is completed by setting \(f_i=\hat{r}_j\)  or \(f_i=\bar{p}_{j}\) .
[Conditional bias of \(\hat{r}_{i}\mid \hat{\lambda }_i\ne \lambda _i\) ]

Affine estimator \(\hat{r}_{i}\)  is not unbiased when conditioned on a function of previous interactions as the last equality of Equation REF  may not hold.

[Covariance of affine estimators]

\(\hat{r}_{i}\)  and \(\hat{r}_{j<i}\)  are uncorrelated as long as \(\hat{\lambda }_i=\lambda _i\) .


By Theorem  and the law of total expectation:

E[rirj]
= E[E[rirjrj]]
= E[rjE[rirj]]
= E[rjE[ri]]
= E[rj]E[ri].
The variance of affine estimator in Equation REF  is:
\(\mathbb {V}[\hat{p}_n] = \frac{1}{(\sum _{i=1}^n \omega _i)^2} \sum _{i=1}^n \frac{\omega _i^2}{\lambda _i^2} ( p(1-p) - (1-\lambda _i)^2 \mathbb {V}[\bar{p}_{i-1}]).\) 
By Lemmas ,  and  :

V[pn]
= 1(i=1n i)2 (i=1n i2V[ri] + 2i=1n-1j=i+1n ijCov[ri, rj])
= 1(i=1n i)2 i=1n i2i2(V [ri] + (1-i)2 V[pi-1] - 2(1-i)
Cov[ri, pi-1])
= (i=1n i)-2 i=1n i2i2 (p-p2 - (1-i)2 V[pi-1]).
 [Convergence of affine estimator]

When \(\forall \hat{\lambda }_i=\lambda _i\) , \(\hat{p}_n=\frac{\sum _{i=1}^n \omega _i \hat{r}_i}{\sum _{i=1}^n \omega _i}\)  converges within \(\epsilon \)  of the true value \(p\)  with probability of at least \(1-\alpha \)  if \((\sum _{i=1}^n \omega _i)^2 \ge \frac{1}{2\epsilon ^2}\log {\frac{2}{\alpha }}\sum _{i=1}^n \big (\frac{\omega _i}{\lambda _i}\big )^2\) .
Define \(X_n\)  as \(\sum _{i=1}^n \omega _i (\hat{r}_{i}-p)\) , \(X_0=0\) . Then \(X_n\)  is a martingale as \(\mathbb {E}[X_n \mid X_0,X_1,...,X_{n-1}]=X_{n-1}+\mathbb {E}[w_i(\hat{r}_i-p)\mid \bar{p}_{n-1}]=X_{n-1}\) . For each \(n: X_n-X_{n-1}=\omega _n (\hat{r}_{n}-p)\) , whose values for \(r_n=0\)  and \(r_n=1\)  are separated by \(d_n=\frac{\omega _n}{\lambda _n}\) . By Azuma's inequality:
\(\begin{split}\text{Pr}(|\dfrac{\sum _{i=1}^n \omega _i (\hat{r}_{i}-p)}{\sum _{i=1}^n \omega _i}| > \epsilon )&=\text{Pr}\bigg (|X_n-X_0| > \epsilon \sum _{i=1}^n \omega _i\bigg ) \\&\le 2\exp (\dfrac{-2\epsilon ^2(\sum _{i=1}^n \omega _i)^2}{\sum _{i=1}^n d_i^2}).\end{split}\) 
The proof is completed by taking the logarithm of both sides and redistributing the terms.
[Convergence of \(\bar{p}_n\)  for specific \(\omega _i\) ]

By Theorem , when \(\forall {}i:\hat{\lambda }_i=\lambda _i\) , \(\omega _i=1\) , \(\hat{p}_n\)  is a consistent estimator of \(p\)  if \(\sum _{i=1}^n\frac{1}{\lambda _i^2}\)  grows slower than \(n^2\) . Similarly, when \(\forall \hat{\lambda }_i=\lambda _i=\omega _i\) , \(\hat{p}_n\)  is a consistent estimator of \(p\)  if \((\sum _{i=1}^n\lambda _i)^2\)  grows quicker than \(n\) .
